The midline is the piece connecting the middle of two of its parties. Respectively, all at a triangle three average lines. Knowing property of the average line and also length of the parties of a triangle and its corners, it is possible to find length of the average line.
It is required to you
- Parties of a triangle, triangle corners
Instruction
1. Let in ABC MN triangle - the average line connecting the middle of the parties of AB (point of M) and AC (a point N). On property the midline connecting the middle of two parties is parallel to the third party and is equal to its half. Means, the average MN line will be parallel to the party of BC and BC/2 is equal. Therefore, for determination of length of a midline it is enough to know length of the party of this third party.
2. Let the parties which middle are connected by the average MN line, that is AB and AC and also BAC corner between them be known now. As MN is the average line, AM = AB/2, and AN = AC/2. Then according to the theorem of cosines fairly: MN^2 = (AM^2)+ (AN^2)-2*AM*AN*cos (BAC) = (AB^2/4)+ (AC^2/4) - AB*AC*cos(BAC)/2. From here, MN = sqrt ((AB^2/4)+ (AC^2/4) - AB*AC*cos(BAC)/2).
3. If the parties of AB and AC are known, then the average MN line can be found, knowing a corner of ABC or ACB. Let, for example, the corner of ABC be known. As on property of the average MN line BC, corners of ABC and AMN - corresponding is parallel, and, therefore, to ABC = to AMN. Then according to the theorem of cosines: AN^2 = AC^2/4 = (AM^2)+ (MN^2)-2*AM*MN*cos (AMN). Therefore, the party of MN can be found from a quadratic equation (MN^2) - AB*MN*cos (ABC) (-AC^2/4) = 0.